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Problems
Contests
International Contests
Mediterranean Mathematics Olympiad
2022 Mediterranean Mathematics Olympiad
2022 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
3
1
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(a + b + c)^2/(a^2+b^2+c^2)+...+ (d+a+b)^5/(d^5+a^5+b^5)<=120
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be four positive real numbers. Prove that
(
a
+
b
+
c
)
2
a
2
+
b
2
+
c
2
+
(
b
+
c
+
d
)
3
b
3
+
c
3
+
d
3
+
(
c
+
d
+
a
)
4
c
4
+
d
4
+
a
4
+
(
d
+
a
+
b
)
5
d
5
+
a
5
+
b
5
≤
120
\frac{(a + b + c)^2}{a^2+b^2+c^2}+\frac{(b + c + d)^3}{b^3+c^3+d^3}+\frac{(c+d+a)^4}{c^4+d^4+a^4}+\frac{(d+a+b)^5}{d^5+a^5+b^5}\le 120
a
2
+
b
2
+
c
2
(
a
+
b
+
c
)
2
+
b
3
+
c
3
+
d
3
(
b
+
c
+
d
)
3
+
c
4
+
d
4
+
a
4
(
c
+
d
+
a
)
4
+
d
5
+
a
5
+
b
5
(
d
+
a
+
b
)
5
≤
120
4
1
Hide problems
PQ tangent to circumcircle of ABC
The triangle
A
B
C
ABC
A
BC
is inscribed in a circle
γ
\gamma
γ
of center
O
O
O
, with
A
B
<
A
C
AB < AC
A
B
<
A
C
. A point
D
D
D
on the angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
and a point
E
E
E
on segment
B
C
BC
BC
satisfy
O
E
OE
OE
is parallel to
A
D
AD
A
D
and
D
E
⊥
B
C
DE \perp BC
D
E
⊥
BC
. Point
K
K
K
lies on the extension line of
E
B
EB
EB
such that
E
A
=
E
K
EA = EK
E
A
=
E
K
. A circle pass through points
A
,
K
,
D
A,K,D
A
,
K
,
D
meets the extension line of
B
C
BC
BC
at point
P
P
P
, and meets the circle of center
O
O
O
at point
Q
≠
A
Q\ne A
Q
=
A
. Prove that the line
P
Q
PQ
PQ
is tangent to the circle
γ
\gamma
γ
.
2
1
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ab222 ... 231 is divisible by 73, cd222... 231 is divisible by 79
(a) Decide whether there exist two decimal digits
a
a
a
and
b
b
b
, such that every integer with decimal representation
a
b
222...231
ab222 ... 231
ab
222...231
is divisible by
73
73
73
. (b) Decide whether there exist two decimal digits
c
c
c
and
d
d
d
, such that every integer with decimal representation
c
d
222...231
cd222... 231
c
d
222...231
is divisible by
79
79
79
.
1
1
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m two-sided cards C_i , with integers from 1-999 on each side
Let
S
=
{
1
,
.
.
.
,
999
}
S = \{1,..., 999\}
S
=
{
1
,
...
,
999
}
. Determine the smallest integer
m
m
m
. for which there exist
m
m
m
two-sided cards
C
1
C_1
C
1
,...,
C
m
C_m
C
m
with the following properties:
∙
\bullet
∙
Every card
C
i
C_i
C
i
has an integer from
S
S
S
on one side and another integer from
S
S
S
on the other side.
∙
\bullet
∙
For all
x
,
y
∈
S
x,y \in S
x
,
y
∈
S
with
x
≠
y
x\ne y
x
=
y
, it is possible to select a card
C
i
C_i
C
i
that shows
x
x
x
on one of its sides and another card
C
j
C_j
C
j
(with
i
≠
j
i \ne j
i
=
j
) that shows
y
y
y
on one of its sides.